# Section 2.1 Matrix Addition, Scalar Multiplication, and Transposition

Definition: 1. For any
$m \times n$ matrix, $A = \begin{bmatrix}\vec{v_{1}} \cdots \vec{v_{n}}\end{bmatrix}$ the i-th entry of $\vec{v_{j}}$ vector is called the (i, j)-entry of

$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n}\\ a_{21} & & & & & a_{2n}\\ \vdots & & & & & \vdots\\ a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in}\\ \vdots & & & & & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mj} & \cdots & a_{mn} \end{bmatrix}$

2. A square matrix is an $n \times n$ matrix.

3. The diagonal entries in an $m \times n$ matrix $A = \begin{bmatrix}\vec{a_{ij}}\end{bmatrix}$ are $a_{11}, a_{22}, \cdots$, and they form the main diagonal of $A$. A diagonal matrix is a square $n \times n$ matrix whose non-diagonal entries are zero

4. The matrix with 1′s on the diagonal and 0′s elsewhere is called an identity matrix and is denoted by I.

5. A zero matrix is a $m \times n$ matrix whose entries are all zero and is written as 0.

6. $A = \begin{bmatrix}\vec{a_{ij}}\end{bmatrix}$ and $B = \begin{bmatrix}\vec{b_{ij}}\end{bmatrix}$ are two $m \times n$ matrices. We say $A$ is equal to $B$ if
$a_{ij} = b_{ij}$ for all $i, j$. The sum of $A$ and $B$ is
$A + B = \begin{bmatrix}\vec{a_{ij}}+\vec{b_{ij}}\end{bmatrix}$.

7. If $r$ is a scalar and $A = \begin{bmatrix}\vec{a_{ij}}\end{bmatrix}$
is a matrix, then the scalar multiple $rA = \begin{bmatrix}\vec{ra_{ij}}\end{bmatrix}$ which entries are $r$ times the entries of $A$.

Note: Only when two matrices of the same size can they be equal. The sum of two matrices is only defined when two matrices are of the same size.

Theorem: $A, B,$ and $C$ are matrices of the same size, and let $r$ and $s$ be scalars.

(a) $A + B = B + A$

(b) $(A + B) + C = A + (B + C)$

(c) $A + 0 = A$

(d) $r(A + B) = rA + rB$

(e) $(r + d)A = rA + sA$

(f) $r(sA) = (rs)A$

Definition: 1. The transpose of $A, A^T$ is the matrix that has rows of $A$ as its columns or has columns of $A$ as its rows.

2. The matrix $A$ is called symmetric if and only if $A = A^T$. Note that this immediately implies that $A$ is a square matrix.

Theorem: Let $A$ and $B$ be matrices whose sizes are appropiate for the sums of products. Then

a. $(A^T)^T = A$

b. $(A + B)^T = A^T + B^T$

c. For any scalar, $r, (rA)^T = r A^T$

Example 1: Verify $(A + B)^T = A^T + B^T$.

Exercise 1: Verify $((A)^T)^T = A$.

Example 2: Find $(2A + B^T)^T$ where $A=\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\end{bmatrix}$ and $B=\begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6\end{bmatrix}$.

Exercise 2: Find $(A^T - 3B)^T$ where $A=\begin{bmatrix} 1 & -2\\ -2 & 0 \\ 3 & 1\end{bmatrix}$ and $B=\begin{bmatrix} 0 & 3 & 4\\ -2 & 1 & 1\end{bmatrix}$.

Group Work 1: Show $A + 2B$ is a diagonal matrix if both $A$ and $B$ are diagonal matrices.

Group Work 2: In each case either show that the statement is true or give an
example showing it is false.

a. If $A + B = A + C$, then $B$ and $C$ have the same size.

b. If $A + B = 0$, then $B = 0$.

c. If the (3,1)-entry of $A$ is 5, then the (1,3)-entry of $A^T$ is 5.

d. $A$ and $A^T$ have the same main diagonal for every matrix $A$.

e. If $B$ is symmetric and $A^T = 3B$, then $A = 3B$.

f. If $A$ and $B$ are symmetric, then $kA + mB$ is symmetric for any scalars $k$ and $m$.

g. $A + A^T$ is symmetric for any square matrix $A$.

h. If $Q + A = A$ holds for every $m \times n$ matrix $A$, then $Q$
is the zero matrix.

Group Work 3: Show $A^T + 3B$ is a symmetric matrix if both $A$ and $B$ are symmetric matrices